Pelosi, as Speaker, is the last step in determining eligibility. She, or the office of the speaker, signs off on the eligibility of the President and VP, before the inauguration. The President, per the rules, cannot control, influence or determine that, it's part of the checks and balances in the Constitution. The power to vet Presidents is by the legislative branch, per the Constitution.

For candidates, the Federal Election Commission is the ruling authority on the matter.

This is just a trap they're trying to drag him into. They want to attack him on it, but he's not biting.